mirror of
https://github.com/vale981/arb
synced 2025-03-06 01:41:39 -05:00
52 lines
1.5 KiB
ReStructuredText
52 lines
1.5 KiB
ReStructuredText
.. _algorithms_hurwitz:
|
|
|
|
Algorithms for the Hurwitz zeta function
|
|
===============================================================================
|
|
|
|
Euler-Maclaurin summation
|
|
-------------------------------------------------------------------------------
|
|
|
|
The Euler-Maclaurin formula allows evaluating the Hurwitz zeta function and
|
|
its derivatives for general complex input. The algorithm is described
|
|
in [Joh2013]_.
|
|
|
|
Parameter Taylor series
|
|
-------------------------------------------------------------------------------
|
|
|
|
To evaluate `\zeta(s,a)` for several nearby parameter values, the following
|
|
Taylor expansion is useful:
|
|
|
|
.. math ::
|
|
|
|
\zeta(s,a+x) = \sum_{k=0}^{\infty} (-x)^k \frac{(s)_k}{k!} \zeta(s+k,a)
|
|
|
|
We assume that `a \ge 1` is real and that `\sigma = \operatorname{re}(s)`
|
|
with `K + \sigma > 1`. The tail is bounded by
|
|
|
|
.. math ::
|
|
|
|
\sum_{k=K}^{\infty} |x|^k \frac{|(s)_k|}{k!} \zeta(\sigma+k,a) \le
|
|
\sum_{k=K}^{\infty}
|
|
|x|^k \frac{|(s)_k|}{k!} \left[
|
|
\frac{1}{a^{\sigma+k}} + \frac{1}{(\sigma+k-1) a^{\sigma+k-1}} \right].
|
|
|
|
Denote the term on the right by `T(k)`. Then
|
|
|
|
.. math ::
|
|
|
|
\left|\frac{T(k+1)}{T(k)}\right| =
|
|
\frac{|x|}{a}
|
|
\frac{(k+\sigma-1)}{(k+\sigma)}
|
|
\frac{(k+\sigma+a)}{(k+\sigma+a-1)}
|
|
\frac{|k+s|}{(k+1)}
|
|
\le
|
|
\frac{|x|}{a}
|
|
\left(1 + \frac{1}{K+\sigma+a-1}\right)
|
|
\left(1 + \frac{|s-1|}{K+1}\right) = C
|
|
|
|
and if `C < 1`,
|
|
|
|
.. math ::
|
|
|
|
\sum_{k=K}^{\infty} T(k) \le \frac{T(K)}{1-C}.
|
|
|